Resonances for perturbations of a semiclassical periodic SchrSdinger operator

Ark. Mat., 32 (1994), 323-371

(~) 1994 by Institut Mittag-Leffler. All rights reserved

Resonances for perturbations of a

semiclassical periodic SchrSdinger operator

Fr@d@ric K l o p p

Abstract. In the semi-classical regime we study the resonances of the operator Pt:--h2A+

V +t. 6V in some small neighborhood of the first spectral band of P0- Here V is a periodic potential, 6V a compactly supported potential and t a small coupling constant. We construct a meromorphic multivalued continuation of the resolvent of Pt, and define the resonances to be the poles of this continuation. We compute these resonances and study the way they turn into eigenvalues when t crosses a certain threshold.

O. I n t r o d u c t i o n

In a previous p a p e r [K1] we studied t h e semi-classical eigenvalue p r o b l e m for t h e following o p e r a t o r acting on L 2 ( R n ) :

(0.1) Pt : - h 2 A + V +t 6V,

where V is a periodic potential, 6V a c o m p a c t l y s u p p o r t e d p o t e n t i a l a n d t a real coupling c o n s t a n t .

For h small e n o u g h we proved t h a t there exists a t h r e s h o l d T~v ( T e v = 0 if n = l or 2, a n d Tsv>O otherwise), such that, for t>T~y, Pt a d m i t s a simple eigenvalue A(t) in a n e i g h b o r h o o d of t h e first b a n d of P0.

In this p a p e r we are dealing w i t h t h e resonance p r o b l e m for Pt, a n d conse- quently, as we will see later on, with w h a t h a p p e n s to A(t) w h e n t gets smaller t h a n T~v.

Resonances have b e e n studied quite extensively for various o p e r a t o r s in t h e last 20 years, a n d there exist various definitions of t h e m , all of which lead t o m a n y investigations (see, for example, t h e works of P. D. L a x a n d R. Phillips [LxPh], J. Aguilar a n d J. M. C o m b e s [AgCo], E. Balslev a n d J. M. C o m b e s [BaCo], M. R e e d a n d B. Simon [ReSi], W . Hunziker [Hul], B. Simon [Sil], [Si2], B. Helffer a n d

324 Fr~d6ric Ktopp

J. Sj6strand [HeSj], B. Helffer a n d A. Martinez [HeMal, P. Hislop a n d I. M . Sigal [HiSig], E. Balslev a n d E. Skibsted [BaSk], A. O r t h [Or], etc.).

However, in m o s t of these papers the authors studied resonances for pertur- bations of the Laplace operator. In our case, w e are interested in resonances for perturbations of a periodic SchrSdinger operator. T h e p r o b l e m of the multivalued m e r o m o r p h i c continuation of the resolvent of an operator of the f o r m (0.i) has already b e e n studied in a quite general setting.

In the one-dimensional case ( n = i), in [Fill, N. E. Firsova constructs the mero- m o r p h i c continuation of the resolvent of Hill operators a n d perturbed Hill operators o n the R i e m a n n surface of quasi-momenta. In [Fi2], she studies resonances for a Hill operator perturbed by an exponentially decreasing potential. She s h o w s that, in each gap (of the Hill operator) of sufficiently high energy, there exists one or an o d d n u m b e r of resonances for the perturbed Hill operator.

In a m o r e general case (no restriction on the dimension), in [G62] (see also [G~I]), C. G 6 r a r d constructs a multivalued m e r o m o r p h i e continuation of the resol- vent of a periodic Schr6dinger operator. H e gives a geometric interpretation of the branch points of this continuation, a n d s h o w s that the branch points contained in a simple b a n d are the critical values of the b a n d function (i.e. the Floquet eigenval- ues). A t last, he also gives s o m e properties of the possible resonances for a periodic SchrSdinger operator perturbed by an exponentially decreasing potential. Never- theless, under such general assumptions, he is not able to s h o w the existence of resonances.

A s w e already pointed out, in this paper w e will study the resonances for

Pt

only in s o m e small n e i g h b o r h o o d of B, the first b a n d of the s p e c t r u m of P0. U n d e r generic assumptions on V this b a n d will be simple for h small enough. Using the reduction d o n e in [Kl], as a first theorem, w e s h o w that

Rt(z)

(resp.

Ro(z))

can be continued as a multivalued m e r o m o r p h i c (resp. analytic) operator-valued function in s o m e small c o m p l e x neighborhood of B. T h e branch points for both of these functions are the critical values of the b a n d function. A n d these branch points are of logarithmic type if n is even, a n d of square root type if n is odd. T o prove these results w e rewrite

Pt

via s o m e adapted Fourier transform a n d w e push the relevant m o m e n t u m space (in our case, it is a torus due to the periodicity of the b a c k g r o u n d potential V ) into C n to m o v e the essential s p e c t r u m a w a y from the real axis (see [G~2]). This is s o m e kind of analytic dilation m e t h o d adapted to periodic problems (see [AgCo], [Hul] and [Cy]).

T h e n we define the resonances as the possible poles of the continuation of the resolvent. We show that, in dimension 1 or in dimension larger t h a n 3, some neighborhood of the interior of B is free from resonances. So we m a y only find resonances near the edges of the band B (at least if n ~ 2 ) .

Resonances for perturbations of a semiclassical periodic Schrbdinger operator 325 If n r and n < 4 (resp. n > 5 ) , we prove the existence of one or more resonances when

t~Tby

(resp.

t<T~y)

and t is close enough to

T~v.

We locate these resonances on the different sheets of the R i e m a n n surface where we could continue

Rt(z).

We compute the asymptotics of these resonances when t tends to

Tbv.

So we can follow the way in which these resonances t u r n into a unique eigenvalue when t increases and crosses

T6v.

In a way these resonances are p e r t u r b e d bound states (see [Hu2]).

Using the previously cited asymptotics, we c o m p u t e the imaginary p a r t of the resonances we found. One must notice t h a t , in odd dimension, there is always one resonance t h a t is located on the real axis of the second sheet of the R i e m a n n surface associated to the problem. If n ~ 3 , it is the only existing resonance (at least in the domain we study).

Finally, using these results on resonances, we prove t h a t there are no eigenvalues for

Pt

embedded in the interior of the b a n d B, at least when t is close enough to

T~v.

T h e p a p e r is organized along the following lines. After this short introduction, in Section 1 we describe our precise framework and state the main results. In Section 2 we construct the analytic continuation of

Rt (z),

the technical p a r t being described in the appendix, Section 4. In Section 3 we c o m p u t e the resonances and prove the absence of embedded eigenvalues.

Acknowledgements.

T h e author would like to t h a n k J. Sjbstrand for suggesting him this subject, as well as B. Helffer, C. Gbrard and L. Robbiano for m a n y helpful discussions, and A. Bardot for efficient typing assistance.

I. D e f i n i t i o n s a n d r e s u l t s 1. D e f i n i t i o n s

L n

Let L be a lattice: =~]~j=l

Zuj

where ( u j ) l < j < n is a basis of R n. Let L*

be the dual lattice of L (i,e L * = { 7 ' e ( R n ) * ; V T e L , 7 ' . T e 2 ~ Z } ) and

T~-(Rn)*/L *,

the dual torus. We consider the following periodic Schrbdinger hamiltonian acting on L2(Rn):

(1.1) P = - h 2 A + V

where

(H.1) V E C ~ ( R n , R ) and V is L-periodic, t h a t is V x e R n, V ~ e L ,

V(x+~)=V(x).

Under assumption (H.1) it is well known t h a t the s p e c t r u m of P consists of bands; these bands consist of purely absolutely continuous spectrum. It is in a

326 Fr@d@ric Klopp

neighborhood of such a band t h a t we are going to define and study the resonances for certain perturbations of P in the semi-classical limit (i.e h--*0).

Let a(P) be the spectrum of P. First, to isolate one of the bands of a ( P ) from the rest of the spectrum, we will need two more assumptions on V. Here we will only give an approximative statement of these assumptions, the rigorous statement being found in [K1]. Suppose

(H.2) for 5 > 0 small enough, {xER~; V(x)<_5} has only one non-empty connected component in each cell of the lattice L; this component is compact and its diameter in the Agmon metric is 0. (The Agmon metric is the metric induced by the measure s u p ( V ( x ) - 5 , O)dx.)

We call the connected components of {xERn; V ( x ) ~ 0 } , the wells of V; by assumption (H.2), these can be indexed by the points of L. For ~,EL we define P~

to be the operator P where all the wells, except the well corresponding to % have been filled (see [K1] for a precise statement).

Let us suppose that there exist #(h), a simple eigenvalue of P0 and a(h), a positive function of h, such that

(H.3)

(i) #(h)---~0, a(h)---+O and hloga(h)---+O when h-+0, (ii) for h small enough,

a(Po)N[p(h)- 2a(h), #(h)+ 2a(h)] = {#(h)}.

One knows that under assumptions (H.1)-(H.3), for h small enough, there exists an analytic function Wh defined in a neighborhood of T in C n, such that, for 0 c T , wh (0) is a simple Floquet eigenvalue for P (i.e. a simple eigenvalue for the operator defined by P on L2={uEL2oc; Vg'cL, u(x+7)=ei~eu(x)} (see [Sj])) and that there is a neighborhood of p(h) of size a(h) in which the spectrum of P consists of the band Wh(T) (see [Or], [K1]).

Let us consider the operator Pt

(1.2) Pt = P + tbV = - h 2 A + V + tSV,

where V satisfies (H.1)-(H.3), t is a real parameter and 5V satisfies

(H.4) 5V is a C a function, compactly supported in a sufficiently small neighbor- hood of the well 0, non-negative and strictly positive in the well 0 (see [K1] for a precise statement).

We now recall the reduction theorem stated in [Zl]. Let Ft(cL2(Rn)) be the spectral space associated to Pt and the interval [ # ( h ) - a ( h ) , #(h)+a(h)] and Ht the orthogonal projection on Ft. Then

Resonances for perturbations of a semiclassical periodic Schrbdinger operator 327

T h e o r e m 1.1.

Assume

(H.1)-(H.4).

(0, ho) and Vte [-a(h)/4, a(h)/4], (a)

There exists some

h0>0

so that Vhc

a(Pt) N

[#(h)-

aa(h), #(h)+

~a(h)] C [#(h)

-a(h), It(h)+a(h)],

(b)

PtHt is unitarily equivalent to ftt:

L2(T)--*L2(T)

defined for

fEL2(T)

by f t t f = Wh" f +b(t)(Ho + K(t) ) f ,

where:

(i)

CO h is the Floquet eigenvalue for P defined above,

(ii) II0

is the orthogonal projection on the vector 1 in

L2(T),

that is, for f E

L2(T)

II0f - Vol (T~ 1/.

f(O) dO,

(iii) K(t): L2(T)-*L2(T)

is an operator whose kernel k(t, O, 0') is analytic in D(O, a(h)/4) x Wh x Wh, where Wh is a neighborhood o f t in C n and D(O, a(h)/4)=

{zeC;

Izl<a(h)/4}. Moreover, there exists

e>0

such that

sup I k(0, 0, 0')I_< e-4h,

W h X W h

sup

I Otk(t, O, 0') I<<_ e -c/h,

D ( O , a ( h ) / 4 ) x W h X W u

(iv)

b(t) is a bi-analytic bijection between two neighborhoods of 0 in C.

Remark.

9 One knows that

b(t) = o.t.

(1

+tq(t))

where q is analytic on D(0,

a(h)/2)

(see [K1] Section 3).

9 In the sequel, for

t e [ - a ( h ) / 4 , a(h)/4],

we will denote by

~ t : F t ~ L 2 ( T )

the unitary equivalence realizing

f~t = ~t Pt IIt ~';.

In fact, 9ct is defined on L2(R n) and K e r ~ t = ( F t ) • The construction of ~ct

([K1] Section 4) shows that, for

t E ( - a ( h ) / 4 , a(h)/4), ~t

and ~'t* can be defined

as bounded operators depending analytically on t from L2(R n) to L2(T) and from

L 2 ( T / t o L2(R n) respectively.

328 Fr6d6ric K l o p p

2. A n a l y t i c c o n t i n u a t i o n o f t h e r e s o l v e n t

The operator

Pt

being self-adjoint, its resolvent ( z - g t ) -1 is well defined as a bounded operator-valued analyticM function of z for z E C \ R . We want to continue analytically

(z-Pt) -1

when z crosses the real axis, z staying in a neighborhood of w(T).

Let tE

[-a(h)/4, a(h)/4].

Then, by definition (1.3)

Pt = ritPtHt +

(1

-IIt)Pt

(1 - H t ) . So for z E C \ R , one has,

(1.4)

(z-Pt) -1 = n t ( z - P t i i t ) - l n t + ( 1 - n t ) ( z - P t ( 1 - n t ) ) - l ( 1 - i i t ) .

For

zEDh={zEC; d(z,

W h ( T ) ) < a ( h ) / 4 } and t E

[-a(h)/4, a(h)/4],

we know by T h e o r e m 1.1 that

(1.5) d(z,

a ( P t (1-Kit))) >

a(h)/4.

So

(1-IIt)(z-Pt(1-IIt))-l(1-IIt)

is a bounded operator-valued analytical func- tion of z in

Dh.

Now, we want to continue analytically

R(z,t)=IIt(z-PtHt)-lIIt

for z in a neighborhood of the band wh(T) when z crosses wh(T). To do this, we will need some assumptions on the Floquet eigenvalue

Wh.

First we give some notations;

we call

Sh=SUPoeTWh(O), ih=infoeWWh(O)

and

f(h)=sh--ih

the supremum, the infimum and the length respectively of the band wh(T). We also renormalize the band defining for 0 E W

wh(0)-ih f(h)

Let us suppose t h a t there exists h0 > 0 such that the following holds:

(H.5)

(i) one has

h.logf(h)

= - S 0 + o ( 1 ) , when h--*0, where So is the shortest Agmon distance between 2 distinct wells.

(ii) there exists W a compact complex neighborhood of T in C n, such that, for hE (0, h0), 02h is analytic in W, the only critical points of 02 h in W are the points of

( ~ L ) / L

and these critical points are non-degenerate.

(iii) there exists C > 0 so that

sup (sup (s p

he(O,ho) \]c~]_<3 \ P E W

Resonances for perturbations of a semiclassical periodic Schr6dinger operator 329

inf ( inf Idet(Hess(wh(0)))0 > C ' he(O,ho) \SE(89

where det(Hess(f(x))) is the determinant of the Hessian matrix of f at the point x.

(iv) for ~>0, there exists 5 ( e ) > 0 so that

VhE(O, ho),lV~h(8)l>5(g ) i f O e W a n d d ( e ,

(~L )/L

1 * 9

)>~.

(v) there exists w0, an analytic function on W, so that uniformly on W, ~ h - " ~ 0

when h-+0.

Remark. Following the appendix of [K1], it can be proven that, under suitable symmetry assumptions on L and V, (H.5) holds.

We define

A0 = {the critical values of ~0} = {~J; 1 ~ j < p } , and

Oj = {the critical points of w0 associated to ~ }

1 * *

={e~;l<k<k,}c (~L)/L.

In the sequel, these critical values will be ordered increasingly with the index j; so

~ = 0 and ~ = i . W e define the internal critical points to be { ~ ; 2 < j _ ~ p - l , i_~

k~_kj}

and the internal critical values to be the critical values associated to these points.

Notice that Wh and wo have the s a m e critical points. Moreover the critical values of wa associated to the points of Oj are tending to ~ w h e n h goes to 0.

W e define

(1.6) R o ( z , t ) = ~ ( Z - - W h ) - l Y t , and

(1.7) Thus (1.8)

r(z, t) = 7,* (H0

+K(t))(z-~h)-if,.

R(z, t) = Ro(z, t ) ( 1 - b ( t ) r ( z ,

t)) -1.

One shows

330 Fr~d@ic Klopp

P r o p o s i t i o n 1.2.

Assume

(H.1)-(H.4).

Then, for h small enough, tE [-a(h)/4, a(h)/4] and z e C \ w h ( T ) ,

(a)

Ro(z, t) is a bounded automorphism of

L2(R n)

satisfying

IIR0(z, t)IIs HR0( z,

t)lls d(z, wh(T))' 1

(b) F(z,t)

is a compact operator from

L2(T)

to

L2(T)

satisfying

Ilr(z,t)ll (.(T <_ 2 f (z_ lh(0))2 dO

Following [G61] and [G~2], we define our set of analytic vectors (i.e. dense subsets of L2(R n) on which we will be able to continue

R(z,t))

to be, for a c R ,

n2a(a n) = {u E :D'(Rn);

e alxl "u e

n2(Rn)}

being provided with its natural norm II~IIL~ =ll eaFI '~IIL:. In fact, L~(R ~) will be a set of analytic vectors only for a > 0 small enough.

Remark.

One shows that there exists C > 0 such that, for h small enough, for

tE[-a(h)/4, a(h)/4]

and for

O<a<l/Ch,

one has

Ht(L2(R~))cL2(Rn)

(see Section 2).

For ( x , r ) E C x R +, define

[3(x,r)

to be a square box in C with center x and side-length 2r and

Dh(X, r)=ih+f(h)E](x, r).

For E c C we define

E • = E n { z 9

C; (Im(z) <> 0) or (Im(z) = 0 and Re(z) ~ ~h(T))}.

Let us define

c[-]h(r0) =

(,wh(W)+f(h)D(O, ro))\

U Q h ( ~ , r 0 ) , l<_j<_p

and ~[~ (ro)= (CDh (ro)) •

In Figure 1, (1) denotes points of wh(Oj), (2) is )~, (3) the length

2ro.f(h)

of the side of the square [2h(A~, r0) denoted by (4). The shaded zones marked by (5) are q2~ (r0).

One has

T h e o r e m 1.3.

Assume

(H.1) (H.5).

Then there exist

h0>0, r0>0

and

c>0

such that for

h 9 h0),

for l <_j<_p, t 9 [-a(h)/4, a(h)/4] and for

zE[:]h~(~, r0),

the following expansions hold:

kj

f(h)Ro(z,t) = ~ S(s

.Hk, 0• (z,-

t)+G~o (s

k = l

Resonances for perturbations of a semiclassical periodic Schr6dinger operator 331

Figure 1.

kj

f(h)r(z, t) = ~ S(2--ah(e])).H~K(2, t)+G~(2, t),

k = l

where:

H + + L(L~(Rn), L2_c(Rn))-valued functions, ann-

(a)

the ( k,o)l<k<_kj and Gk, o are

lyric in

(2,

t) for

(2, t)EO(~, r0) • D(0,

a(h)/4),

H i +

C(L2c(Rn),n~(R~))-valued

functions, ann-

(b)

t h e ( k,K)l<_k<_kj and G g are

lytic in

(2,

t) for (~, t)eO(~Jo, to) • D(O, a(h)/4),

(c)

if n is odd,

T:. ( _ 1 ) ( n _ l ) / 2 z ( n _ 2 ) / 2 ' S ( z ) =

if n is even~

1 (_l)n/2z(n_2)/2.10g

Z.

S ( z ) = 5"

332 Frederic Klopp

Moreover, Ro(z, t) (resp. F(z, t)) can be analytically continued from COh~(ro) to

~Oh(rO) as an s L2_~) (resp. C(L 2, L2))-valued analytic function in z and t.

Remark. s F) is the set of bounded operators from E to F , C(E, F) the set of compact ones. Here z 1/2 and log z are the principal determinations of the square root and the logarithm.

For F a simply connected domain in C and E c F a domain, blC(E, F) denotes the universal covering of E in F. For r o > 0 and l<_j<_p, we define

o)LU )

Let Ro e and F • be the analytic continuations of Ro and F defined from Ohi(), j , r 0 ) x D ( 0 , a(h)/4) to btC(ro,j) x n ( 0 , a(h)/4), and from

COih (ro) xD(O,a(h)/4) to COh(ro) x D ( 0 , a(h)/4).

Using (1.8) one immediately gets the following

C o r o l l a r y 1.4. Assume (H.1)-(H.5). Then there exists h0>0, r 0 > 0 and c > 0 such that, for hE(O, ho) and l < j < p ,

9 R can be meromorphically continued from

D h• 0,-j r0) x D(O,a(h)/4) to blC(ro,j)xD(O,a(h)/4) as R i , an/:(L 2, L2_c)-valued meromorphic function in z and t.

9 R can be meromorphically continued from

cO~(r0) x D ( 0 , a(h)/4) to cOh(ro) x D ( 0 , a(h)/4) as R • an f~(L2c, L2_c)-valued meromorphic function in z and t.

Notice that, when one is dealing with the continuations at the edges of the band (i.e. j = l or p), continuation from above the band or below is the same.

3. R e s o n a n c e s We define

Res• (r0, t) = {z E c Oh(r0); z is a pole of R+(z, t)}, and, for OcLtC(ro,j),

Res• t ) = {z e O; z is a pole of

R+(z, t)}.

Resonances for perturbations of a semiclassical periodic Schr6dinger operator 333 D e f i n i t i o n . z is said to be a resonance for Pt if there exists r o > 0 such that

z E R e s + ( r 0 , t ) , or

3 1 < j < p, 3 (9 C blC(ro, j) such that z E Res + ((9, t).

Remark. In this definition, no difference is m a d e between actual eigenvalues and resonances.

U C ( r o , j ) is naturally provided with a riemannian metric (induced by the eu- clidian metric on C). For 0 < r < r o , let O ( r , j ) be the ball of radius r around A~ in LtC(ro,j).

Our first t h e o r e m states t h a t there are no resonances in a neighborhood of the interior of the band a~h(T).

T h e o r e m 1.5.

(a) For any dimension n there exist h o > 0 , r 0 > 0 such that, for hE(0, h0) and tE ( - a ( h ) / 4 , a(h)/4) one has

Res + (r0, t) = r

(b) For n > 3 and 2 < _ j < p - 1 , there exists h0>0, r 0 > 0 such that, for hE(0, h0) and tE ( - a ( h ) / 4 , a(h)/4) one has

Res + (O(ro, j), t) = r

Remark. In dimension n = 2 we get no results near the internal critical values.

In dimension n = 1 such critical values do not exist.

So, if Pt admits resonances, these arc located near the edges of the b a n d w(T).

We will only study what h a p p e n s near the upper edge of the band; obviously, a s y m m e t r i c s t u d y m a y be done near the lower edge. To c o m p u t e these resonances we will need one more assumption on the b a n d function Wh (an assumption t h a t is satisfied under suitable s y m m e t r y conditions on L and V when h is small enough (see [K1] Appendix)); we assume t h a t

(H.6) there exists only one critical point Os E ( 89 such t h a t w h ( O s ) = s = ~ is the m a x i m u m of co h on W.

We define

D~ = I det(Hess(~h(Os) ) )1-1/2.

We know t h a t near the edges of the band, R + = R - so, for any t and O, a sufficiently small neighborhood of the edges of the band, one has

Res+(O, t) : R e s - ( O , t) = Res(O, t).

334 Frederic Klopp

Under assumption (H.6), we define the following realisation of

14C(ro, Sh)=

14C(ro, p).

Define

[]p (r0,

Sh ) = 8h +e ipH" (Dh (rO,

0)\ [--to

. f ( h ),

0]),

UC(ro, s~) = {sh}U U Dp(ro, s~).

pEZ We define for q E N

UC(r0,sh,q)=(sh}U U Gp(ro,sh).

--q~_p~_q

We are now able to state the results about resonances near the upper end of the band Wh(T). In dimension 1 we get

T h e o r e m 1.6.

Let

n = l

and assume

(H.1)-(H.6).

Then there exists

ho>0, t o > 0 , t o > 0

such that, for hE(O, ho), there exist:

(i)

a function A:(O, go.f(h))--~(s,s+ro.f(h)) the values of which are simple eigenvalues of Pt and that admits the following convergent expansion:

A(t)--Sh

= f ( h ) ' ( t ) 2" ( E a l ( h ) ' ( t ) l ) ,

\ l E N /

where:

9 g = t / f ( h )

9 for any

IEN, a z ( h ) E R

~ 21/2.TrO.Ds ) 2

C~o(h) = \ Vol(T)

.(l+O(e-c/h)) for some

c > O ,

(ii)

a function ~: (-to.f(h),O)-+i+e2i~r.(O, ro.f(h)) the values of which are simple resonances of Pt and that admits the following convergent expansion:

)~(t)--sh =e2i~r" f ( h ) ' ( t )2" (l~N(~l(h)'(t)l ) 9 Moreover, for t E ( - t o ' f (h),

0)U(0, t o ' f (h)),

one has

Res(UC(r0, sh), t) = {~(t)}.

Remark.

For t > 0 , .~(t) is the eigenvalue already found in [K1]. One can notice that for any t, Im(A(t))=0; so the resonances we get for t < 0 are 0-energy resonances.

Resonances for perturbations of a semiclassical periodic SchrSdinger operator 335

l ~ - - ~ l h ~ _ + t > 0 ~ _ + ~---- S l h - ~ 0 Figure 2a.

t < 0 t . . . . x - - - ) 1

8h J

Figure 2b.

Figure 2 shows the resonance picture near both edges of the band. The sheet numbered 0 is the physical sheet and number 1 the non-physical sheet. Figure 2a shows the picture of the resonances when t>0; we omitted to draw the (resp. non-) physical sheet near the upper (resp. lower) edge of the band as it contains neither eigenvalue nor resonance. Figure 2b shows the picture of the resonances when t < 0 . In Figure 2, X denotes an eigenvalue and x a resonance. 0 is the physical sheet and 1 the non-physical one. For t < 0 or t > 0 , we only drew the sheet where there is either an eigenvalue or a resonance. The + or - signs indicate how the sheets are connected.

In dimension 2 we get

T h e o r e m 1.7. Let n = 2 and assume (H.1)-(H.6). Let q E N . Then there exist hq>O, tq>0, rq>O such that there exists a function

),: (0, t q . f ( h ) ) --+ (Sh, sh +rq" f ( h ) )

the values of which are simple eigenvalues for Pc and that admits the following convergent expansion:

so(h)

~(t)-s=/(h).exp ~ ~ Z

l ~ _ O , m ~ _ O

\

m _ . ~ l

s~,m(h).(t)~ m e mao(h)/~),

where:

9 t = t / f ( h )

9 for any (l,m), SZ,m(h)ER

Vol(T). (l+O(e_C/h))

s 0 ( h ) - 2QTI.D8 for some c > O.

336 Frederic Klopp

- 2

t ... - 1

[ . . . X - . . . . 8h

[

... - ~

+1

Figure 3.

Moreover, for re(0, {q . f ( h ) ), one has

Res(UC(rq,

sh, q), t) = {~(t)},

and, for te(-{q, f(h), 0),

Res(blC(rq, Sh, q), t) = r

Remark. A is the eigenvalue already found in [K1]. So, in dimension 2, one does not get any actual resonance near the edges of the band but only eigenvalues (see Figure 3). One may draw a symmetric picture for t<0.

In Figure 3, X is an eigenvalue for t>0. The numbers at the left of the sheets indicate how these are connected.

Resonances for perturbations of a semiclassical periodic SchrSdinger operator 337 Let n_> 3. Define

1 / T 1

I = Vol(T----~"

sh--wh(e) de.

Let T~v be the threshold for the existence of an eigenvalue outside the band Wh (T) (see [K1]). Then one has

T h e o r e m 1.8. Let n = 3 and assume (H.1)-(H.6). Then there exist h o > 0 , r 0 > 0 , t o > 0 such that, for hE(O, ho), there exist:

(i) a function A:(Tav, Tsv +to. f ( h ) )--~Sh +(O, t o ' f (h)) that is a simple eigen- value of Pt and that admits the following convergent expansion:

\ l E N /

where:

9 t = ( t - T ~ y ) / f ( h ) 9 for any I c N , a z ( h ) E R

{ V o l ( T ) . ~ . ( f ( h ) . I ) 2 ~

2.

(l+O(e-c/h)), for some c

>

O.

a0(h) =

\

25/21r2.D8 ]

(ii) a function )~: (Tsy-{o" f(h), T6y]-+Sh + e 2i~" (0, ro" f ( h ) ) that is a simple resonance of Pt and that admits the following convergent expansion:

\ f E N

Moreover, for tE (Tev-{o. f(h), Tsv +{o" f(h)), one has Res(UC(r0, sh), t) = {~(t)}.

Remark. Here one sees that, for decreasing t, the eigenvalue turns into a res- onance when t crosses the threshold Tsv. The resonance satisfies Im(A(t))=0; it is located on the real axis of the second sheet of the Riemann surface where the resolvent ( z - P t ) -1 is defined (see Figure 4).

In Figure 4, X is an eigenvalue for t>T~v and x a resonance for t<_Tsv.

To describe the resonances in dimension 4 we will need

L e m m a 1.9. Let qEN. There exist rq, rq and ! rq~>0, a a function analytic in a small neighborhood of (0, 0) in C 2 that is real if both of its arguments are real and such that c~(z, z')=o(Izl+lz'l) , and, for each - q < j < q , an open set Dj satisfying

2~j~ , ^" " " " e 2~j~.(D(o, r'q)\i.

e . ~ - q , r ~ c , j c (-r'~,0]),

338 Fr4ddric Klopp

<

... X ...

8h

< I ... -x ... - > I

Figure 4.

and such that, if for zE1)j we define

- z ( (log(]ogl~g z )

yj(z) : 177gz" 1 + ~

1))

' l o g z '

then yj: l)j--+e 2ij€ (D(O, rq) \ i. ( - r q ,

0])

is bijective and

- y ~ ( z ) . l o g ( y j ( z ) ) = z .

We then get

Theorem 1.10.

Let q E N and assume

(H.1)-(H.6).

Then there exist hq>O, rq

>0, tq >0

such that, for hE

(0,

hq), there exists an analytic function g admitting the following convergent expansion near (0,

0):

g(v,w)= ~ ~,z(h).v~.w m,

l>_O,m>_O

where

9 for any

(l,m), a,~,z(h)ea

c~0,0(h) =

V ~ "(l+O(e-~/h)), for some c > O,

4re 2 "Ds

such that if we define [= t-T~v

f ( h ) and A p ( t ) - s = f(h).yp(e2iq~f).g({, yp(e2iq~{)/{), for -q<p<_q and t e ( ( T e y - t q . f ( h ) , T e y +tq. f ( h ) ) , then

9 for t > T e y , Ao(t) is a simple eigenvalue of Pt, 9 fort<_T5y, Ao(t) is a simple resonance of Pt,

9 forpT~O and for any t, i v ( t ) is a simple resonance of Pt.

Resonances for perturbations of a semiclassical periodic SchrSdinger operator

Moreover, for t e ( T~y - tq . f ( h ) , Tsy + tq . f ( h ) ) , one has Res(UC(rq, sh, q ) , t ) = U

{Ap(t)}.

--q~_p~_q

339

and for

zED(O, r q ) \ i . ( - r q , O)

Vol(T)

{89

7r. (-1)(n-1)/2z(n-4)/2

if n is odd,

g n ( z ) = 1 ( ~n/2

~ . ~ - l j .z (~-a)/2.1og

z if n is even.

Then one has

T h e o r e m 1.11.

Let n>_5, q C N and assume

(H.1)-(H.6).

Then there exist

hq>0, rq>0, tq>0

such that, for hE(O, hq), there exists an analytic function an admitting the following convergent expansion near

(0, 0):

Z 'vz'wm,

l~_O,m~_O where

9 for any (1, rn),

c~,m(h)ER

9 (~'~,o(h)=--(~.I2)/OI.(l+O(e-c/h)),

2 n ~ / 2 (QI2)(n-2)/2 .Ds.(l+O(e-C/h)), OZ n

0,1(h)---- F ( n / 2 ) . V o l ( T ) . f ( h ) 2 ( - O I ) n / 2 for some

c>O

such that if we define

{= ( t - T ~ v ) / f ( h ) and )~p(t) - s = e 2iwr. ( t - T s y ) ' a n ({, gn

(e2W~{)),

for -q<_p<_q and t E ( T ~ y - { q , f ( h ) , T e v +{q. f(h)), then

9 for t>_T5v, Ao(t) is a simple eigenvalue of Pt, 9 for t < T e v , Ao(t) is a simple resonance of Pt,

9 f o r p r and for any t, Av(t) is a simple resonance of Pt.

9 ( S h _ W h ( O ) ) ~ dO,

Remark.

We compute the imaginary part of these resonances and get, for - q <

p<_q,

p.Tr.{

Im()~p(t))=.o,o(h). f ( h ) .

((l~g ~ ' ) . ( 1 +o(1)).

Figure 5b shows a picture of these resonances.

Let n > 5 and define

340 Frederic Klopp

. . . X . . .

8h

X

X

[ ... X .... - ~ 1

Figure 5a. n_~4, n odd.

Moreover, for t e ( Tsv - tq . / ( h ) , T~v + tq " f ( h ) ) , one has

Res(UC(rq, sh,q),t) = [_J

- - q ~ p ~ q

Remark. O n e c o m p u t e s t h e i m a g i n a r y p a r t of these resonances to obtain:

9 if n is even, for -q<p<_q,

Im(Ap (t)) = ~.p. f ( h ) . a~, 1 (h). ( - 1) n/2. (t)(~-2)/2, 9 if n is odd, for p even,

Im(Ap(t)) = 0 and, for p odd,

Im(Ap(t)) = f ( h ) . a ~ , l ( h ) . ( - 1 ) (p+1)/2 9 (n-2)/2 Figure 5a, b, c, show pictures of these resonances d e p e n d i n g on n.

In all these pictures, x is a resonance for t < T ~ v (or t<_T~v if n = 4 ) a n d X is an eigenvalue or a resonance for t>_Tsv (or t > T ~ y if n = 4 ) d e p e n d i n g on in which sheet it is located.

4. E m b e d d e d e i g e n v a l u e s

A corollary of t h e preceding s t u d y is

Resonances for perturbations of a semiclassical periodic Schr6dinger operator 341

~ 3 2 X

t . . . x . . . - 2

~21

^{x t .... X}

... ^{- 1}

] . . . X . . .

Sh

t-- ... X . ~ +1

Figure 5b. n_>4, n_----0(4).

T h e o r e m 1.12.

Let n E N and assume

(H.1)-(H.6).

Then there exist

h0>0

and

t0>0

such that, for

hE(0, h0),

one has

(a)

if n = l : f o r t E [ - t o . f ( h ) , t o . f ( h ) ] , there is no eigenvalue of Pt embedded in [ih, 8hi, the band of the essential spectrum, that is

(r( Pt ) N [ih, 8h ] =

O'cont

( Pt ) N [ih, 8h ].

(b)

if n>3: for t E [ T b v - t o . f ( h ) , T b v + t o . f ( h ) ] , there is no eigenvalue of Pt embedded in (ih, Sh), the band of the essential spectrum, that is

ff(Pt)N(ih, 8h) : (Tcont(Pt)N(ih, 8h).

Remark.

In dimension 2 one can state a similar result outside some neighbor-

hoods of the inner critical points of

Wh.

342 Frederic Klopp

- 2

C x t ... x ...

_{- 1}

C

1

8! . . . x . . .

Figure 5c. n_>4, n--2(4).

II. A n a l y t i c c o n t i n u a t i o n o f Ro a n d F 1. T h e u n i t a r y e q u i v a l e n c e Jc't

We will first recall some facts from [K1]. Under assumptions (H.1)-(H.4), let

(~t,~)~eL

be the Hilbert basis spanning Ft constructed in [K1]. We know that there exist h0>0 and C > 0 such that, for hE(0, h0),

tE(-a(h)/4, a(h)/4)

and for any

~EL,

(2.1)

II~t,~(')el'-~l/ChliL2(Rn ) <_ C.

One defines the projector Ht on

Ft,

for p E L 2 ( R n ) ,

(2.2) IItp =

E(~l~t,,v)~t,,v,

~'EL

Resonances for perturbations of a semiclassicM periodic SchrSdinger operator 343

where (.I.) denotes the scalar product in L2(Rn).

We also define 9vt: L2(Rn)--*L2(T) and .T'~:L2(T)--~L2(R n) for ~oEL2(R~), uEL2(T) and 0ET by

(2.3) (.Ttp)(O) = ~-~'~ (~ I ~,,~)~,~.o

^,

~EL

and

(2.4) f't*u= ~ ( V o l T ) f r e-i'Y'~ u(O)dO) qot''~"

~t realises a unitary equivalence from Ft to L2(T) with inverse ft*.

For a>0, let Wa=T+iBc~(O,a) and O(Wa) be the set of bounded analytic functions in W~ provided with the L ~ norm (here Bc~ (0, a) denotes the ball of center 0 and radius a in cn). One has

L e m m a 2.1. There exists C0>0 and h0>0 such that for hE(O, ho) and 0<

a<a' <l/Coh

(a) IIt is continuous from L2a(a n) to L2a(a n) ,

(b) Yzt is compact from L2a(R n) to O(Wa,), (c) 5rt * is continuous from O(Wa) to L],(R~).

These results are uniform in t and h small enough.

Proof. Let ~EL](Rn). Then for 7EL

(2.5) (~ I ~,~) = L~ (e-(aix'+lx-~'/ch))" (~a,xl~(X)). (elX-~'/C%~,~ (X)) dx

so by (2.1) (2.6)

=Ce--~

Then, using (2.2) and taking Co large one gets

"/EL

which proves (a).

344 Fr~dgric Klopp

Using estimate (2.6), (b) is immediate. To prove (c) one uses Stokes' formula and the analyticity of u to write, for 7 5 0

T e-eY'~ u(O) dO = /T-i(~"9/Iq e-i'Y'~ u(O) dO f -i-~.o / "a~/'~ dO

_<

Vol(T).e -~'lq IIq oO,Wo.

Then, using (2.4), one

^gets

(c).

2. T h e analytic continuation of Ro and r

By (1.6) (1.7) and (2.3)-(2.4), for ~EL2(R n) one has ( 1 f;e-i"/'~

( 2 . 7 ) R~ Vof(T) z-~h(O) dO ~,~

"yEL a n d

(2.s) r(z't)(~)="~[(j[T (l+k(t'"O))'~'t(cp)(O) dO .

Let O<a'<a<l/Ch. For 7EL we define II~(z): (.9(Wa)--~C by

1

s e"~~

u ~ Vol(T---5"

z-~h(O) ~0 and rk(z, t):O(Wa)--~O(Wa) by

1 fT (l+k(t,., 0)).u(O)

(2.9) u H Vol(T~" Z--wh(O) dO.

B o t h of these operators are compact.

After renormalizing the b a n d by introducing s w e prove, using Proposition A.I, that the operators

f(h)II~(z)

^{a n d}

f(h)Fk(z,t)

can be analyti- cally continued from above or below the real axis to a neighborhood of w h ( T ) \

{Wh(O]); l~_j~_p, l~k~kj}

as c o m p a c t operators from

O(Wa)

to C or

O(Wa)

with the following upper b o u n d s for the norm:

f(h) lln~(z)llo(wa)~c < CeN"Im(s

Resonances for perturbations of a semiclassical periodic Schr6dinger operator 345

a n d

f(h)IIrk(z, t)HO(Wo)-~o(wo) <_ c

for s o m e C > 0.

Moreover, Proposition A.2 gives us precise expansions for these continuations near the branch points which are the critical points of Wh. Using this, one easily gets T h e o r e m 1.3 by (2.7) and (2.8).

Estimate (b) of the Proposition A.2 applied to

OtFk(z,t)

c o m b i n e d with the estimates k n o w n for the kernel k implies that, for (~,t)E[Z(~, r 0 ) •

a(h)/4),

one has

(2.10) E

IIO~H~,k(z, t)II

:E N

o(wo)~O(Wo)+ Ila~Ck (z, t)II O(Wo)---,O(Wo)

• ~ _<

e_c/h.

l < / < k j

I I I . T h e s p e c t r u m o f r

By (2.8) and (2.9) F and Fk are unitarily equivalent. So we will now study the spectrum of Fk. Taking z and t as usual, let us define F0(z):

(D(Wa)---+(9(Wa)

by

1 fT ~(0)

u H Vol(W~

Z--Wh(O) dO,

where the expression to the right is viewed as a constant function. Obviously, the spectrum of F0 consists of 2 eigenvalues, 0 of infinite multiplicity and

• = 1 fT 1

Vol(W) "

Z--Wh(O--~ dO

of multiplicity I with eigenvector the constant function 1. For

xr

one computes

0

(2.11) (x-r~ = x

By Proposition A.2 one has an expansion kj

f (h)F0 (z) -- E S ( s Wh (O~)). g z t 0 (s + G~ (s

/=1

Then using (2.11) one gets the estimate

1 cIs(z)l (2.12) II(x-ro(z))-lll-< r ~ +

Ixl.lx-I(z) l

Ixl

346 Frederic Klopp

Using estimate (b) of Proposition A.2 applied to F k - F o , for (s j,r0) x D(0,

a(h)/4),

one gets

(2.13) E

l<l<kj II H~k (~, t)-H,~o (~)IIo(wo)-~o(wo)

+11C~(2, t)-C~o (2)llo(wo)-~o(wo) < e-~lh.

1. Investigations near the regular values of Wh

One gets

L e m m a 2.2.

For any

ro>0

small enough there exist

ho>0

and

C > 0

such that, for

he(0, h0)

and for zel4C(C[3(ro), ~

E]+(ro)),

a(rk(z, t)) C {0,

I ( z ) } + D c (0, C.

IS(z)l.

Ilklloo,wo).

Proof.

By Propositions A.1 and A.3 we know that, for ro small enough, zE b/C(r c rq• and for some C > 0 (depending only on to),

1 C

(2.14)

C. f(h---~ <-

]I(z)] <

f(h---~"

By Proposition A.1 we know that, for

zEl4C(CE](ro), ~

[3+(ro)) and some C>0,

1 Z k(t,.,O') dO'

Ilrk(z,t)-ro(z)llo(wo) <_

V o l ( T ~ _ .

Z--Wh(O') O(Wa)

< C'llkll~,wo

- f(h)

By (2.12) and (2.14) weget that,

ifxr

for some

C > 0 and h small enough, then

II(x-Po(Z))-lllo(wa) <

So using the following resolvent formula (2.15)

f(h)

C.llkll~,Wo

( x - r k(z, t)) -1 = (x-ro(Z)) -1. ( 1 - ( r k(Z, t)-I~o(z)) .(x-cO(Z, t)) -1)-1,

w e get the a n n o u n c e d l e m m a .

Resonances for perturbations of a semiclassical periodic Schrhdinger operator 347

2. I n v e s t i g a t i o n s n e a r t h e c r i t i c a l v a l u e s

Let

1<jKp

and let

)~Yo=ih+f(h)"~Jo

be a critical value of~0 rescaled to the size of the band. We recall that, for 0 < r < r 0 , O ( r , j ) is a ball of radius

r.f(h)

around )~o in

ldC(ro,j).

Then the following holds:

L e m m a 2.3.

Assume n>_3. There exist hr>O,

C r > 0

when O<r is small such that, for hE(O, hr) and zCO(r,j), one has

a(G(z, t)) c {O,I(z)}+Dc(O, G.II(z)l.llkll ,Wo).

Proof.

If n > 3 , the proof is exactly the same as for the case of the regular values except that one has (2.14) only for z in the usual complex plane (the first sheet of the universal covering). So the upper bound on

I(z)

still holds true for z equal to a branch point (as S is continuous at z = 0 ) . T h e n by connectedness of the universal covering and continuity of

I(z),

one gets this bound in a neighborhood of the branch points in the universal covering. The same is true for the estimate of

IIFk(z, t ) - F 0 ( z ) l I. This gives the lemma.

In dimension n = l or 2 we will assume (H.6). We will only study the spectrum of Fk near the edges of the band. For q C N and

Z=Sh

or

ih

define

OC(q,r,z)= U (z+e P 'Dh(r'O))cOC(r'z)"

--q~p~q One has

L e m m a 2.4.

Assume

n = l

or 2. For any qEN, there exist ha>O , rq>O and

C q > 0

such that, for he(O, hq) and zEO(q, rq,Sh)UO(q, ra,ih) one has a(G(z, t))c

D c (0,

~h))UDc(I(z), Cq'lI(z)l'HkH~,wo).

Remark.

Notice t h a t when 2 tend to

~h, f(h).I(z)

tends to co, so for 2--~h chosen small enough,

Dc (0, ~ h ) ) NDc(I(z), Cq.,I(z)l.,,k,,~,wo) = r

This holds uniformly in h.

Proof.

We will only prove L e m m a 2.4 for z close to the maximum; the case of the minimum goes along the same lines. Using the fact t h a t there is only one

348 Fr@d@ric Klopp

critical point corresponding to the maximum, say 0~, by the expansion given in Theorem 1.3 and the computations done in Section 4 one has, for z close to

Sh,

f(h).Fk(z,

t) =

S(5-gh)Ak +a(5, t).

Here

G(z,t)

is a compact operator such t h a t for some

Cq>O

and

re>O ,

for z in (PC(q,

rq, Sh)

and t as usual

Ila( ',

t)llo(wo) O(Wo) <

G.

Moreover, Ak is the rank one operator defined for

uEO(Wa)

by

Ak(u)(O)

=/7. (1 + k ( t , O, Os)).u(O~), where

1 2n/2.Vol(OB(O,

1)).

Idet(Hess(~h(Os)))n -1/2

_</3 = Vol(W)

for some C > 0 independent of h small enough.

By perturbation theory we know that

_<C

(2.16)

cr(f(h).Vk(z,

t)) C D c ( 0 ,

Cq)UDc

(~'S(Y--~h), Cq. 19"S(~--~h)l"

Ilkll ,w ).

By Proposition A.2 we know that there exists

C a

> 0 such that for

zEO(q, rq, Sh)

1 I(z).f(h) I

- -

So using (2.16) one gets the announced result.

Remark.

1. In odd dimension, because the branch points are of square root type (so the Riemann surface associated to the analytic continuation is only finitely

"sheeted"), one may choose

rq, hq,

and Cq independent of q.

2. Here we did not treat the case of the inner critical points in dimension n = 2 (such points do not exist in dimension n = l ) . In dimension 2, one sees that, when h goes to 0, for any 0 < k < k j ,

~h(0 k) tends

to ),% Hence, for 2 tending to one of the Wh(0]), it is not possible to control the behaviour of the expansions given in Proposition A.2 without further assumptions on the behaviour of the critical points.

As will be seen in the next section, these lemmas will suffice to conclude that resonances can only exist near the edges of the band. So we are going now to study more precisely the spectrum of Fk near

Sh

(the other side of the band can be treated in the same way).

R e s o n a n c e s for p e r t u r b a t i o n s of a semiclassical periodic S c h r S d i n g e r o p e r a t o r 349

Proposition 2.5. Assume (H.6) and let qEN. Then there exist hq>O, rq>O

and Cq>O such that, for he(O, hq), there exists a function vp(z,k) defined on OC(q, rq, Sh) verifying:

(a) vp(z, k) is a simple eigenvalue of Fk and

a(Fk(z, t) )NDc (I(z), II(z)---~i ) = {vp(z,k ) }.

(b) For zEOC(q, rq, Sh) we define the coefficients a~,mER by

f(h)'I(z) = S(z--sh)" (Ea~ ') + E a~ 1 (Z--Sh)l'

\ / E N : 1EN

and al~ if m>_2.

Then, for zEOC(q, rq, Sh ), the function vp(z, k) admits the following uniformly convergent expansion:

If n = 1 or 2 then

f(h).vp(z,k) ~-S(2--gh). E a~, m(t)(~-~h)l'(S(z-sh))-m'

I~N,mEN

where the coefficients (a~,m(t))zeN,,~eN are analytic functions in rEDo(0, a(h)/4), real valued for t real and

lakt,m(t)-a~m I < Cq.r~ 1.S(rq) m. Ilkll~,w~,

IO~a~.(t) I <_ C~ .r( * 9 S ( ~ ) ". IIkll~,~:o.

I] n>_3 then

f(h).vp(z,k)= ~ a~,m(t)(2--~h)Z.(S(~--gh)) m,

IEN,mEN

where the coefficients (a~,m(t))leN,meN are analytic functions in reDo(0, a(h)/4), real valued for t real and

]a~,l (t)-a~,ol <_ Cq.rq~.S(rq)-m.}lk}loo, w~, la~,o(t)-a?,ll < Cq.r; z.s(rq) -~.llklloc,wo, {a~.~(t) I <_ Cq .r~ z .S(r~) -m. IIklloo,wo

10ta~,m(t)l < Cq .r; t. S(r~) -"~. [[kll~,w~

for m > 2,

for any m.

350 Pr6d6ric Klopp

Remark.

9 The remark following Lemina 2.4 still holds for this proposition.

9 Using the notations of appendix A one has aO 1 AO(1 ) (4zr) n/2

o,o = Vol(T) r ( ~ / 2 ) . V o l ( T )

]dct(Hess(wh(O~)))[-U2'

if 1 < n < 2, and

aO 1 (4~r) "/2

o,1 - Vol(T) "A~ = F ( n / 2 ) . V o l ( T ) '

Idet(Hess(wh(O~)))[-1/2'

if n > 3.

Moreover, for n > 3 and _{- -}

0<l<(n-3)/2, a ~

_{l , O} is given by the expansion (A.17) for J(~, 1), that is

a~

s

^(-1)~

~,0 = (~h_~h(O))~+ ~ dO.

Proof.

For h small enough consider the following family of projectors

IIk,~ = fc(X-r~.k(z,t))-ldx, a ~ [0,1],

where C is the complex contour {xEC;

Ix-I(z)l=-II(z)l/4}

and F~.k is the operator Fk where one has replaced the kernel k by a . k . This family is analytic in a in the norm sense. By (2.12), (2.15), and by chosing h small enough to get

]]kl[~,w o < 88

we see that

Hk,o

is of rank 1. So IIk,1 is of rank 1 which gives point (a).

Assume n = l or 2. For

yEC

and

5~Dc(sh, rq),

consider the operator

where

Hk(2, t)

and

Gk(5, t)

are the operators given in Theorem 1.3 (in this

case,

because one looks at the edge of the band, these operators do not depend on whether one continues analytically from below or above the band). Then Theorem 1.3 says, that for

zEldC(q,

re, s~)

(2.17)

f(h)'Pk(z,t) = S(5--Sh)'Ok ( s(~l~h) ,z).

Noticing that

lim ( sup 1

~-~Okzeotq,r,~h)l S(~-~h) ) ^{= 0 ,}

R e s o n a n c e s for p e r t u r b a t i o n s of a s e m i c l a s s i c a l p e r i o d i c S c h r h d i n g e r o p e r a t o r 351

and letting Z---+Sh in (2.17), one gets

9 1

By part (a) of this proposition, Ok(0, Sh) admits a simple eigenvalue in D c ( 1 , 88 isolated from the rest of its spectrum.

As

Ok(y,2)

is analytic in y, 5 and t, there exists b>0 such that, for

(y,2, t)E

Dc(0, b) x

Dc(gh,

b) x Dc(0,

a(h)/4),

there exists

v(y, 2, k(t)),

a simple eigenvalue of

Ok(Y, z)

isolated from the rest of the spectrum. This eigenvalue is simple and therefore analytic in its parameters, so it admits the following convergent expansion

m,

1CN,mCN

where the coefficients

a~m(t )

are analytic functions in t t h a t are real when t is

k 0

real. The estimates on

lal,m-al,ml

^and

IOta~,,~l

are immediate consequences of the Cauchy estimates applied to this expansion and the estimates (2.10) and (2.13).

By (2.17), if

1/S(2-~h)EDc(O,b),

then

~ ~ 1

This gives the convergent expansion for

vp(z, k(t))

in the case n = l or 2.

Assume n > 3 and let Hk(2, t) and G(2, t) denote the same operators as above.

For y E C and 2EDc(~h,

rq)

consider the operator

Ok(y, 2) = y.Hk(2, t)+G(~,

^t).

For the same reasons as above there exists b>0 such t h a t for (y,

2, t)EDc

^{(0, b)x}

Dc(gh, b) x Dc(0,

a(h)/4)

there exists

v(y, 2,

k(t)), a simple eigenvalue of

Ok(y, 2)

isolated from the rest of the spectrum. Moreover,

v(S(2--gh), 2, k(t)) = f(h).vp(z, kit)).

Now the conclusion follows along the same lines as in the case n = 1 or 2.

I I I . C o m p u t a t i o n o f t h e r e s o n a n c e s

By (1.8) and Corollary 1.4, to say that z is a pole of R + ( z , t ) is equivalent to say t h a t 1 is an eigenvalue of

b(t)F• t).

352 Frederic Klopp

1. I n v e s t i g a t i o n s a w a y f r o m t h e e d g e s o f t h e b a n d

(a)

Far away from the internal critical values.

Let r0 be small enough and h0 be chosen as in L e m m a 2.2. T h e n there exists a constant C ( r 0 ) > 0 such that, for

zel~C(CD(ro),

[3• one has

IZ(z)l < C(r0)

- f ( h ) "

By Proposition A.3 we know t h a t there exists c ( r 0 ) > 0 such t h a t

z 9 uc(cD(r0), D •

when

I Im(I(z))l>_c(ro)/f(h).

So, by Lemma 2.2 and by the estimate on Ilkll~,wo given in Theorem 1.1 one has for

z9

[ ] e ( r 0 ) ) r q R and h small enough

(3.1)

a(Fk(z,t))

C

D(O, e--1/Ch)u Z 9 C; ]Im(z)l >_ ~ e(r0) } ,

for a certain C > 0. But, for t E [ - a ( h ) / 4 ,

a(h)/4],

we know that

b(t)

E R and Ib(t) l <

2a(h). Then, by (3.1), for h small enough, 1 can not be an eigenvalue for

b(t).

Fk(z, t), so z can not be a resonance of

Pt.

(b)

Close to the internal critical values for n>_3.

Except for the fact that one uses L e m m a 2.3 instead of L e m m a 2.2 the proof is the same as above.

Remark.

In dimension 1, the only critical values are the extrema. In dimen- sion 2, the problem near the internal critical values (i.e. that are no extrema) comes from the fact that if there are 2 critical values of ~d h t h a t are asymptotically equal when h goes to 0, then in the expansions given for

I(z)

near these values there may occur compensations for I m ( I ( z ) ) (i.e. this imaginary part may become 0).

Consequently, the largest eigenvalue of

b(t).Fk(z,

t) may be real and equal to 1 (see the remark following L e m m a 2.4).

2. C o m p u t a t i o n o f t h e r e s o n a n c e s n e a r t h e e d g e s o f t h e b a n d

We will only study what happens in a neighborhood of

Sh,

the maximum of Wh.

(a)

Proof of Theorem

1.6. Let

n=

1. Using L e m m a 2.4 and the expansion given in Theorem 1.1 for b(t), we see that for

te[-ro.f(h),ro.f(h)]

(for a certain r 0 > 0 given by L e m m a 2.4), the only eigenvalues of

b(t).r~(z,

t) t h a t may be equal to 1

Resonances for p e r t u r b a t i o n s of a semiclassicM periodic Schrhdinger operator 353

are the ones contained in

Dc(b(t).I(z),

[b(t).I(z)[/4). So by Proposition 2.5 we just have to solve

(3.2)

b(t).vp(z, t) = 1.

Let us first make a change of variables. Let

t=t/f(h).

Then

b(t)

= ~.~. ( l + f ( h ) .

(t.q(tf(h)))) = g.t.

( l + ~ ( t ) ) ,

f(h)

where ~ is analytic in D c ( 0 ,

a(h)/f(h))

and satisfies [~[

<C.f(h).

By Proposition 2.5 and the remark following it, we know that there exists r0 > 0 such that for z in b/C(r0,

Sh)

f(h).vp(z,k) =S(2--~h). E a~, m(t)(2-~h)t'(S(2-sh))-m"

tEN,mEN

Therefore (3.2) becomes

(3.3) g.t.(l+~(t')).S(,5--,~h). E

ate, "~(f(h)'~)(2-~h)t'(S(2-~h))-m=l"

tEN,mEN

For z in

LtC(ro, Sh)

set

(3.4) 2--~h = ~ 2

where fiEDc(0, r0).

Doing this, we uniformize the function S on

LtC(ro, Sh).

Plug (3.4) into (3.3) and use the definition of S to get

(3.5)

~'Q. .(l+~(t)).Ea~,,~(t.f(h)).- ~

.u2t+m = 1.

2 l,m

For t ~ 0 and ~ 0 , (3.5) becomes

(3.6) g(t, f i ) = 0 ,

where

(3.7) 7 ~ " ~ ~ m

l ) m

354 Frederic Klopp

It is clear that g is analytic in a n e i g h b o r h o o d of (0, 0), a n d that

(3.8) g(0,0) = 0 and

Oeg(O,O) = 1.

We can apply the implicit function theorem to equation (3.6) in a neighborhood of (0,0) in C 2. So there exist t0>0, r 0 > 0 and an analytic function ~:Dc(0, t0)--+

Dc(0, r0) such that, for t e D c ( 0 , t0),

=0.

Moreover, w e c o m p u t e

(3.9)

2 where ~ is function analytic in De(0, t0).

As the coefficients

a~,,~(t)

are real for real t, g(t,~) is real when t and ~ are real. Hence the coefficients of the power series expansions of ~ and ~ are real. The estimate of the leading coefficient of ~([) comes from (3.6), (A.16) and Proposi- tion 2.5.

Plugging (3.9) into (3.4) w h e n / 5 0 , we get the announced result. Of course, one can do a symmetric study in

LtC(ro,iu). []

Proof of Theorem

1.7. Let n = 2 . By the same arguments as in the proof of Theorem 1.6, it is clear that we only have to solve equation (3.2) for

vp(z,

t), the eigenvalue of

Fk(Z, t)

given by Proposition 2.5.

Let qCN. Using the expansion of

vp(z,t)

given by Proposition 2.5, for zE

LtC(q, rq, Sh),

we have to solve equation (3.3). Again we will uniformize S on

l~tC(ro, Sh),

the change of variable now being

(3.1o)

z - s h = exp( ),

where ~ E ( - c x ~ , - R 0 ) + i R for some R0>0. For some R e > 0 , exp is an analytic embedding from ( - o c ,

-Rq)+i(-q.lr+~r/2,

q.Tr+Tr/2) into

LtC(q, rq, Sh).

Plug (3.10) into (3.3) and use the definition of S to get

.~-m.exp(/.g) = 1.

(3.11) 2 l,m

Let ~ = l / g . Notice that, for

fLE(-oc,-Rq)+i(-q.~r+Tr/2, q.lr+Tr/2),

one has R e ~ < 0 . For t r and ~r

(3.11)

becomes

(3.12)

R e s o n a n c e s for p e r t u r b a t i o n s of a semiclassical periodic Schr6dinger o p e r a t o r 355

where

/,m m 1

(3.13) g(t,~)=-~.t.(l+O(t)).Ea~,m(t'f(h))" (-~)'~'~ . e x p ( ~ ) - ~ . Obviously g can be defined as a function with continuous derivatives in some neigh- borhood of (0,0) in C• and then it satisfies (3.8). So we can apply the implicit function theorem to get a unique solution to equation (3.12) which, moreover, is of the form

(3.14) ~(t)= ~'a~176 "t" (l+~(t)).

2

The condition Re(~(t))<0 tells us that there is no solution of (3.12) when t<0. To get some precision on ~, we look for solutions to (3.12) of the form

~(~)-- 0"a~176 (0) .~. (1 +~.~(~))-1, 2

where ~ is a function such that ~(0)=C, a constant to be chosen later on. Plug this ansatz into (3.13) to get

g(t' Q'a~176 "t'(l-Ft'w)-l)

~ Q'aok,o(_0)'t~ m

=--~-"t'(l+q(t))'Ea~"~(t'f(h))" \ 4 . ( l + t . ~ ) ]

(3.15) 2 z,.~

• -F o'a~176

\ ~.ao,o(O).t 2 . ( l + t . ~ ) So, for t~0, (3.12) becomes

(3.16) where

(3.17)

h (t, ~ 'exp ( - ~.a0k,20(0).~) ' ~ ) =0'

h(t, ~, ~) : t.f(t, ~, ~) = a~176 (0) + ( l + q ( t ) ) . E a~,,~(t.f(h))

l + t . ~

_/,rn

ao o!0! /

4.(l+t. w)] .t.x.exp,Q~o,o~,)/.[~J

356 Fr6d@ric Klopp

The function f is analytic in some neighborhood of (0, 0, 0) in C 3 and satisfies /(0,0, 0) = O~h(O, O, O)

(3.18) =

C.a~,o(O )

o'a~176176 4-a0k0(0).0~(0)+f(h).0tak0,0(0) 4

and

O~f(O,O,O)-=--ako,o(O).(1

~'a41 ( 0 ) ) .

Choose C such that f(0, 0, 0)=0. Using (A.16) and the estimates on a k given in _l,m Proposition 2.5, we can apply the implicit function theorem uniformly for h small enough to construct N c C 2, a neighborhood of (0,0), and an analytic function

~: N--*C such that f(t, ~:, z~(t, 2 ) ) = 0 . Then by (3.16), we get that 9, the solution of (3.12), satisfies

.0))

2 ~ Q.a0k0(0)

Using the properties of the coefficient

a~,m(O )

one completes the proof of Theo- rem 1.7. []

Proof of Theorem

1.8. Let n = 3 . Let t 0 > 0 be fixed and arbitrarily large. We know that

I(z)/f(h)

is bounded in

lAC(ro, Sh)

for a certain r0>0. By Lemma 2.3 and using the fact that

Ilkllo~,w~ <e -c/h

(for a certain c>0), it is clear that, for h small enough (depending on t0) and

te[-to.f(h),to.f(h)],

the only eigenval- ues of

b(t).rk(z,t)

that may be equal to 1 are the ones contained in

Dc(b(t).

I(z), Ib(t).I(z)I/4).

We only have to solve equation (3.2) for

vp(z,

t), the eigenvalue of Fk(z, t) given by Proposition 2.5. We recall that by Proposition 2.5 there exists r 0 > 0 such that, for z in 5/C(r0,

Sh),

IEN,mEN

Z

where the properties of the coefficients

akl,m

are given in Proposition 2.5.

Letting z--~Sh, we see that we need to take t > 0 for (3.2) to admit a solution in UC(r0,

Sh)

for r 0 > 0 small enough. (3.2) can be rewritten

(3.19) )lJ0m=l

/EN,mEN